Advances in Time-Domain Computational Electromagnetic Methods E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: IEEE/OUP Series on Electromagnetic Wave Theory (formerly IEEE only), Series Editor: Donald G. Dudley.
• Sprache: Englisch

Seitenzahl: 958

Veröffentlichungsjahr: 2022

Beschreibung

Advances in Time-Domain Computational Electromagnetic Methods

Discover state-of-the-art time domain electromagnetic modeling and simulation algorithms

Advances in Time-Domain Computational Electromagnetic Methods delivers a thorough exploration of recent developments in time domain computational methods for solving complex electromagnetic problems. The book discusses the main time domain computational electromagnetics techniques, including finite-difference time domain (FDTD), finite-element time domain (FETD), discontinuous Galerkin time domain (DGTD), time domain integral equation (TDIE), and other methods in electromagnetic, multiphysics modeling and simulation, and antenna designs.

The book bridges the gap between academic research and real engineering applications by comprehensively surveying the full picture of current state-of-the-art time domain electromagnetic simulation techniques. Among other topics, it offers readers discussions of automatic load balancing schemes for DG-FETD/SETD methods and convolution quadrature time domain integral equation methods for electromagnetic scattering.

Advances in Time-Domain Computational Electromagnetic Methods also includes:

• Introductions to cylindrical, spherical, and symplectic FDTD, as well as FDTD for metasurfaces with GSTC and FDTD for nonlinear metasurfaces
• Explorations of FETD for dispersive and nonlinear media and SETD-DDM for periodic/ quasi-periodic arrays
• Discussions of TDIE, including explicit marching-on-in-time solvers for second-kind time domain integral equations, TD-SIE DDM, and convolution quadrature time domain integral equation methods for electromagnetic scattering
• Treatments of deep learning, including time domain electromagnetic forward and inverse modeling using a differentiable programming platform

Ideal for undergraduate and graduate students studying the design and development of various kinds of communication systems, as well as professionals working in these fields, Advances in Time-Domain Computational Electromagnetic Methods is also an invaluable resource for those taking advanced graduate courses in computational electromagnetic methods and simulation techniques.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

About the Editors

List of Contributors

Preface

Part I: Time-Domain Methods for Analyzing Nonlinear Phenomena

1 Integration of Nonlinear Circuit Elements into FDTD Method Formulation

1.1 Introduction

1.2 FDTD Updating Equations for Nonlinear Elements

1.3 FDTD–SPICE

1.4 Data-Based Models

1.5 Conclusions

References

2 FDTD Method for Nonlinear Metasurface Analysis

2.1 Introduction to Nonlinear Metasurface

2.2 Fundamentals of Classical Models

2.3 FDTD Analysis

2.4 Applications

2.5 Summary

References

Notes

3 The Finite-Element Time-Domain Method for Dispersive and Nonlinear Media

3.1 Background and Motivation

3.2 Dispersive and Nonlinear Media

3.3 Finite-Element Time-Domain Formulations

3.4 FETD for Dispersive and Nonlinear Media

3.5 Stability Analysis

3.6 Conclusion

References

Note

Part II: Time-Domain Methods for Multiphysics and Multiscale Modeling

4 Discontinuous Galerkin Time-Domain Method in Electromagnetics: From Nanostructure Simulations to Multiphysics Implementations

4.1 Introduction to the Discontinuous Galerkin Time-Domain Method

4.2 Application of the DGTD Method to Real Problems

References

5 Adaptive Discontinuous Galerkin Time-Domain Method for the Modeling and Simulation of Electromagnetic and Multiphysics Problems

5.1 Introduction

5.2 Nodal Discontinuous Galerkin Time-Domain Method

5.3 Modeling and Simulation of Electromagnetic–Plasma Interaction

5.4 Dynamic Adaptation Algorithm

5.5 Multirate Time Integration Technique

5.6 Numerical Examples

5.7 Conclusion

References

6 DGTD Method for Periodic and Quasi-Periodic Structures

6.1 Introduction

6.2 The Subdomain-Level DGTD Method

6.3 Memory-Efficient DGTD Method for Periodic Structures

6.4 Memory-Efficient DGTD Method for Quasi-Periodic Structures

6.5 Conclusions

References

Part III: Time-Domain Integral Equation Methods for Scattering Analysis

7 Explicit Marching-on-in-time Solvers for Second-kind Time Domain Integral Equations

7.1 Introduction

7.2 TD-MFIE and Its Discretization

7.3 TD-MFVIE and Its Discretization Using FLC Basis Functions

7.4 Predictor–Corrector Scheme

7.5 Implicit MOT Scheme

7.6 Comparison of Implicit and Explicit Solutions

7.7 Computational Complexity Analysis

7.8 Remarks

7.9 Numerical Results

7.10 Conclusion

References

Note

8 Convolution Quadrature Time Domain Integral Equation Methods for Electromagnetic Scattering

8.1 Introduction

8.2 Background and Notations

8.3 Solution Using Convolution Quadrature

8.4 Implementation Details

8.5 Acceleration, Preconditioning, and Stabilizations

8.6 Details of the Numerical Examples Used in the Chapter

8.7 Conclusions

8.A List of RK Methods

References

9 Solving Electromagnetic Scattering Problems Using Impulse Responses

9.1 Introduction

9.2 Impulse Responses

9.3 Behavior at the Interior Resonance Frequencies

9.4 Impact on MOT Late Time Instability

9.5 Analytical Expressions for the Retarded-Time Potentials

9.6 Numerical Verification of Stability Properties

9.7 Effect of Impulse Response Truncation

9.8 Domain Decomposition Method Based on Impulse Responses

9.9 Conclusions

References

Part IV: Applications of Deep Learning in Time-Domain Methods

10 Time-Domain Electromagnetic Forward and Inverse Modeling Using a Differentiable Programming Platform

10.1 Introduction

10.2 RNN-Based Formulation of Wave Propagation

10.3 Gradient Sensitivity Analysis

10.4 Electromagnetic Data Inversion Fulfilled by Network Training

10.5 Conclusion

References

11 Machine Learning Application for Modeling and Design Optimization of High Frequency Structures

11.1 Introduction

11.2 Background

11.3 Applications of Machine Learning to Electromagnetics

11.4 Discussion

References

Part V: Parallel Computation Schemes for Time-Domain Methods

12 Acceleration of FDTD Code Using MATLAB's Parallel Computing Toolbox

12.1 Introduction

12.2 Parallelization with MATLAB

12.3 Multi-CPU and Multi-GPU for FDTD Simulation

12.4 Sample Results

12.5 Conclusions

References

13 Parallel Subdomain-Level Discontinuous Galerkin Time Domain Method

13.1 Introduction

13.2 Comparison of Parallel Element- and Subdomain-Level DGTD Methods

13.3 Parallelization Scheme for Subdomain-Level DGTD Methods

13.4 Application of the Parallel Subdomain-Level DGTD Method for Large-Scale Cases

13.5 Conclusion

References

14 Alternate Parallelization Strategies for FETD Formulations

14.1 Background and Motivation

14.2 Challenges in FETD Parallelization

14.3 Gaussian Belief Propagation for Solving Linear Systems

14.4 Finite Element Formulation of Gaussian Belief Propagation

14.5 Parallelization of Nonlinear Problems

14.6 Implementation on Parallel Hardware

14.7 Conclusion

References

Note

Part VI: Multidisciplinary Explorations of Time-Domain Methods

15 The Symplectic FDTD Method for Maxwell and Schrödinger Equations

15.1 Introduction

15.2 Basic Theory for the Symplectic FDTD Method

15.3 The Coupled Maxwell–Schrödinger Equations

15.4 A Unified Symplectic Framework for Maxwell–Schrödinger Equations

15.5 Numerical Simulation

15.6 Conclusion

Acknowledgments

Author Biography

References

16 Cylindrical FDTD Formulation for Low Frequency Applications

16.1 Cylindrical Finite-Difference Time-Domain Method

16.2 Convolutional PML in Cylindrical Coordinates

16.3 Cylindrical FDTD Formulation for Circuit Elements

16.4 Concluding Remarks

References

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Description of numerical constants for the Gummel–Poon BJT model....

Table 1.2 Description of numerical constants for the MESFET model.

Chapter 2

Table 2.1 SHG conversion efficiencies.

Chapter 3

Table 3.1 Coefficients of the polynomials for the VWE FETD with

.

Table 3.2 Coefficients of the polynomials for the VWE FETD with

.

Chapter 4

Table 4.1 Length scales.

Table 4.2 Time scales.

Table 4.3 Physical parameters used for the PCD examples.

Chapter 6

Table 6.1 Simulation time and memory consumption by the memory-efficient DGT...

Table 6.2 Simulation time and memory used by conventional subdomain-level DG...

Table 6.3 Resource consumption of the conventional subdomain-level DGTD meth...

Table 6.4 Simulation time and memory consumption of quasi-periodic scatters....

Table 6.5 Simulation time and memory consumption of the quasi-periodic patch...

Table 6.6 Resource consumption of the conventional subdomain-level DGTD meth...

Chapter 7

Table 7.1 Comparison of the time costs

a)

of different stages of both MOT sch...

Table 7.2 Comparison of

,

, and

at

for

and

.

Table 7.3 Comparison of the time costs of both MOT solvers.

Table 7.4 Computation times required by all four MOT schemes for three sets ...

Table 7.5 Execution times and RCS errors of

for all three simulations.

Table 7.6 Execution times required by all four MOT solvers for three sets of...

Chapter 10

Table 10.1 Computational costs of computing 20,000 time steps on PyTorch and...

Table 10.2 Computational costs of gradient computation with AD and the finit...

Table 10.3 Normalized model misfit with the gradient descent method using di...

Table 10.4 Normalized model misfit with the Momentum method using different ...

Table 10.5 Normalized model misfit with the RMSprop method using different l...

Table 10.6 Normalized model misfit with the Adam method using different lean...

Chapter 13

Table 13.1 The time overhead of 3000 time steps for interface integration.

Table 13.2 The relationship between the 3000 communications and DoFs of inte...

Chapter 14

Table 14.1 Numerical values of model parameters.

Chapter 15

Table 15.1 The coefficients

c

l

 = 

c

...

Table 15.2 The coefficients for different-order accurate collocated differen...

List of Illustrations

Chapter 1

Figure 1.1 Improved model of junction diode incorporating junction and diffu...

Figure 1.2 Schematic symbol and small-signal model of bipolar junction trans...

Figure 1.3 Ebers–Moll model of a bipolar junction transistor.

Figure 1.4 Gummel–Poon model of a bipolar junction transistor.

Figure 1.5 Field-effect transistor model.

Figure 1.6 Nonlinear FET model.

Figure 1.7 Schematic illustration of FDTD–SPICE integration.

Chapter 2

Figure 2.1 The trend for metasurface research: from “technology-based” schem...

Figure 2.2 Different approaches for material modeling.

Figure 2.3 Lagrangian description for classical approach. The oscillator equ...

Figure 2.4 Intuitive illustrations for two most popular quantum concepts now...

Figure 2.5 A simple but straightforward explanation for probability variatio...

Figure 2.6 The basic steps of deriving Maxwell-hydrodynamic model from Boltz...

Figure 2.7 System diagram of the FDTD-TDPM algorithm.

Figure 2.8 Computational grids in FDTD algorithm. (a) Standard Yee's grid. (...

Figure 2.9 Flow charts of the FDTD-TDPM algorithm. (a) Flow chart of the gen...

Figure 2.10 Surface model for metallic material surfaces. Surface A is the e...

Figure 2.11 The element geometries and unit cell setup of nonlinear metasurf...

Figure 2.12 Linear responses for different kinds of element structures. The ...

Figure 2.13 Nonlinear responses for different kinds of element structures. T...

Figure 2.14 All-optical switch (AOS) configuration: (a) AOS geometry, and (b...

Figure 2.15 Simulation results for the AOS by FDTD-TDPM algorithm.

Figure 2.16 Two-dimensional dispersion diagram for understanding the behavio...

Figure 2.17 Normalized nonlinear responses for extracting the equivalent sec...

Figure 2.18 Harmonic-modulated NMS (HM-NMS) configuration.

Figure 2.19 Quantum interpretations for the generalized phase conjugation (P...

Figure 2.20 Simulation results for HM-NMS with versatile functions. Left col...

Chapter 3

Figure 3.1 Plane wave normally incident on an infinitely large dispersive di...

Figure 3.2 Numerical and exact values of the reflection and transmission coe...

Figure 3.3 Demonstration of the effects of anomalous linear dispersion in a ...

Figure 3.4 Creation of a temporal soliton via the introduction of dielectric...

Figure 3.5 Numerical and exact values of the reflection and transmission coe...

Figure 3.6 A coaxial-fed cylindrical monopole antenna placed on a ground pla...

Figure 3.7 Reflected voltage at the terminal of the cylindrical monopole ant...

Figure 3.8 Normalized electric field recorded inside the PML over 10 million...

Figure 3.9 Diffuse interference pattern resulting from diffraction in a bulk...

Figure 3.10 Creation of a focused spatial soliton due to the presence of an ...

Figure 3.11 Linear (a) and logarithmic (b) visual depiction of the spectral ...

Figure 3.12 Linear (top) and logarithmic (bottom) plots of the pulse's norma...

Chapter 4

Figure 4.1 Illustration of the scatterer, computation domain boundary

, Huy...

Figure 4.2 The illustration of temporal interpolation process.

Figure 4.3 Cluster of three PEC spheres. All dimensions are in meters. Repro...

Figure 4.4 Transient scattered electric field computed at the origin of clus...

Figure 4.5 RCS on (a)

- and (b)

-planes computed at 2.53 MHz from the solu...

Figure 4.6 RCS on (a)

- and (b)

-planes computed at 1.003 GHz from the sol...

Figure 4.7 U-shaped scatterer. All dimensions are in meters.

Figure 4.8 The bistatic (a) and monostatic (b) RCS of the PEC U-shaped scatt...

Figure 4.9 The bistatic (a) and monostatic (b) RCS of the dielectric U-shape...

Figure 4.10 The calculated RCS in

(a) and

(b) planes with the DGTDBI met...

Figure 4.11 Fitted magnitude and phase of the surface conductivity

from 50...

Figure 4.12 The calculated magnitudes of coefficients

,

, and

by the pro...

Figure 4.13 Magnitude of calculated transmission coefficient

by the propos...

Figure 4.14 Normalized ECS versus frequency for a freestanding graphene patc...

Figure 4.15 TSCS and ACS of the

freestanding graphene patch corresponding...

Figure 4.16 Calculated forward RCS of the

freestanding graphene patch corr...

Figure 4.17 Normalized near-field distribution of

over the graphene sheet ...

Figure 4.18 Normalized far-field pattern in

- and

-plane at

for

.

Figure 4.19 (a) Total transmission coefficients, (b) Faraday rotation angle,...

Figure 4.20 Calculated ECS by the proposed DGTD-RBC algorithm and the refere...

Figure 4.21 (a) Normalized total scattering cross-section (TSCS) and (b) ECS...

Figure 4.22 Normalized ECS for substrates Si,

, and

.

Figure 4.23 Time marching scheme.

Figure 4.24 Cross-section of the conventional PCD considered in the first ex...

Figure 4.25 (a)

computed using the DG scheme at different time instants. T...

Figure 4.26 (a) The time signature and (b) the spectrum of the optical EM wa...

Figure 4.27 Cross-section of the plasmonic PCD.

Figure 4.28 (a)

computed using the DG scheme at different time instants. T...

Figure 4.29 (a) The time signature and (b) the spectrum of the THz current g...

Chapter 5

Figure 5.1 Optimized interpolating nodes in tetrahedron [9]. (a)

p = 6

...

Figure 5.2 (a) Spectral radius of the facial matrix versus polynomial basis ...

Figure 5.3 Illustration of an adaptively refined nonconformal grid.

Figure 5.4 Snapshots of the

y

-component of the electric field and the corres...

Figure 5.5 Electric field distributions at (a) 360, (b) 460, and (c) 560 ns ...

Figure 5.6 Comparison of the electric field distribution at 460 ns along the...

Figure 5.7 Comparison of CPU time consumptions among the DGTD simulations us...

Figure 5.8 Geometry model of a parallel-plate waveguide with two metallic in...

Figure 5.9 Effective field distribution at (a) 0.284, (b) 0.404, and (c) 0.6...

Figure 5.10 Zoomed-in view of the electron density distribution around the i...

Figure 5.11 Comparison of the cumulative computational time when the uniform...

Figure 5.12 Comparison of the numerical solutions recorded at the center poi...

Figure 5.13 Illustration of the problem setting with two conducting cylinder...

Figure 5.14 Investigation of the dynamically adjusted polynomial order distr...

Figure 5.15 The first row: the

y

-component of the secondary electric field o...

Figure 5.16 Electron density distribution observed at (a) 0.25; (b) 0.50; (c...

Figure 5.17 Distribution and evolution of the electron density (solid line) ...

Figure 5.18 Spacing of the fully developed plasma filaments at 2.00 ns.

Figure 5.19 Propagation of the plasma front recorded at different plasma den...

Chapter 6

Figure 6.1 Application examples of periodic/quasi-periodic structures. (a) A...

Figure 6.2 Truncation effect of a finite array. (a) The frequency selected s...

Figure 6.3 Schematic of domain decomposition in the subdomain-level DGTD met...

Figure 6.4 Typical finite period structures.

Figure 6.5 One volume and six surface integrations of a cell.

Figure 6.6 Embedded model of a cell in the periodic structure.

Figure 6.7 PEC resonant cavity with (a) 16 cells, (b) 32 cells, (c) 48 cells...

Figure 6.8 The

x

component of the electric field recorded at the receiver fo...

Figure 6.9 Memory consumption for the cavities with different number of cell...

Figure 6.10 Patch antenna array. (a) Overview. (b) Unit cell. (c) Side view ...

Figure 6.11 The

S

-parameters of the ports in the patch antenna array using t...

Figure 6.12 The

S

-parameters of the patch antenna array using the ImEx-RK ti...

Figure 6.13 Example of the quasi-periodic structure.

Figure 6.14 PEC cavity filled with quasi-periodic structures. (a) Geometric ...

Figure 6.15 The

x

components of the electric fields.

Figure 6.16 Quasi-periodic patch antenna array. (a) Overview. (b) Specific d...

Figure 6.17 Three-dimensional radiation patterns of the quasi-periodic patch...

Figure 6.18 Two-dimensional radiation patterns of the quasi-periodic patch a...

Figure 6.19 Structure of the quasi-periodic antenna array. (a) Overview. (b)...

Figure 6.20 The scattering parameters of the quasi-periodic patch antenna ar...

Chapter 7

Figure 7.1 Solution coefficient obtained by the explicit and the implicit MO...

Figure 7.2 Comparison of

,

, and

for

,

, and

.

Figure 7.3

and

versus

. Reproduced with permission from [34].

Figure 7.4 Comparison of the coefficient of an RWG basis function using the ...

Figure 7.5 Snapshots of the surface current at times (a)

, (b)

, and (c)

Figure 7.6 Comparison of

and

to

at

for

and

.

Figure 7.7 (a)

and (b)

computed by the implicit and explicit MOT schemes...

Figure 7.8

,

, and

versus

for

at

.

Figure 7.9

and

versus

for

.

Figure 7.10 (a) Comparison of

computed using all four MOT schemes at

for...

Figure 7.11

computed at

using all four MOT schemes for (a)

, (b)

, and...

Figure 7.12 (a)

calculated using the explicit MOT scheme for TD-MFVIE with...

Figure 7.13 (a)

obtained at

from all four MOT solvers for the simulation...

Chapter 8

Figure 8.1 Sequence of transform for the convolution quadrature solution of ...

Figure 8.2 Norm of the interaction matrices in the differentiated EFIE for d...

Figure 8.3 Current density on the surface of a sphere of radius 1 m illumina...

Figure 8.4 Current density on the surface of a sphere of radius 1 m illumina...

Figure 8.5 Eigenvalues of

for the stabilized EFIE on a sphere. All the eig...

Figure 8.6 Eigenvalues of

for the differentiated EFIE on a sphere. The clu...

Figure 8.7 Eigenvalues of

for the MFIE on a sphere. All the eigenvalues ar...

Figure 8.8 Current density on the surface of a torus of outer radius 1.5 m a...

Figure 8.9 Surface current density induced on a shuttle model at different t...

Chapter 9

Figure 9.1 Transient impulse responses and their spectra. (a) (a1) TD-CFIE, ...

Figure 9.2 Static current patterns in the impulse response of the TD-EFIE. T...

Figure 9.3 Scattering problem of a PEC scatterer.

Figure 9.4 (a) Circuit model for interior resonance associated with EFIE. (b...

Figure 9.5 The numerical results in the vicinity of

ka = υ01

...

Figure 9.6 The numerical results in the vicinity of

ka = μ11

...

Figure 9.7 Geometric parameter definitions. (a) Observation point

r

projecti...

Figure 9.8 Sampling scheme for the temporal convolution. The shadowed triang...

Figure 9.9 PEC sphere. Solutions to TD-EFIE, TD-MFIE, and TD CFIE. (a) Wide-...

Figure 9.10 Eigenvalues of the TD-IEs [21]. (a) TD-EFIE. (b)TD-MFIE. (c)TD-C...

Figure 9.11 Early time solutions to the TD-CFIE with different time steps [2...

Figure 9.12 PEC almond. (a) Late time responses in logarithm. (b) Early time...

Figure 9.13 The TD-CFIE solutions by truncated impulse responses [21]. (a) E...

Figure 9.14 A scatterer enclosed by the Huygens surface.

Figure 9.15 Total field imposed on the

m

-th block in a multi-module system....

Figure 9.16 The configuration for the almond array. (a) Single model. (b) Ar...

Figure 9.17 RCS results of the

3 × 3

almond array. (a) Sing...

Chapter 10

Figure 10.1 RNN model demonstration. (a) Compact form. (b) Unfolded form.

Figure 10.2 Single cell architecture of RNN model for simulating wave propag...

Figure 10.3 RNN model for simulating wave propagation using FDTD.

Figure 10.4 Simulation results by PyTorch and Matlab. (a) From Receiver 1. (...

Figure 10.5 The true model. (a) The true distribution of permittivity. (b) T...

Figure 10.6 Gradient computed by AD.

Figure 10.7 Gradients computed by the finite-difference method with differen...

Figure 10.8 Inversion results based on the gradient descent method with diff...

Figure 10.9 Inversion results based on Momentum method with different learni...

Figure 10.10 Inversion results based on RMSprop method with different learni...

Figure 10.11 Inversion results based on the Adam method with different learn...

Chapter 11

Figure 11.1 A feedforward shallow ANN. The outputs from the hidden layer

V

d...

Figure 11.2 A multi-layer feedforward deep neural network.

Figure 11.3 The ReLU activation function.

Figure 11.4 An illustration of a convolution layer. A 3 × 3 filte...

Figure 11.5 A max pooling operation applied to a two-dimensional input.

Figure 11.6 An illustration of two layers of a DNN. A convolutional layer wi...

Figure 11.7 A recurrent neural network.

Figure 11.8 An illustration of the drop out step. Neurons (gray color) are r...

Figure 11.9 The flow sequence of a machine learning problem.

Figure 11.10 Relacing a time-intensive electromagnetic simulation with a tra...

Figure 11.11 An illustration of the forward ANN approach. The values of the ...

Figure 11.12 An inverse ANN is used to map the responses to the correspondin...

Figure 11.13 An illustration of the approach presented in [62]; the real and...

Figure 11.14 The neuro-space mapping approach: (a) input neuro-space mapping...

Figure 11.15 An illustration of an Image-based ANN. An image of a metasurfac...

Figure 11.16 A metasurface presented by VO

2

cold and hot pixels on a MgF

2

su...

Figure 11.17 The reflection from the VO

2

metasurface calculated using COMSOL...

Figure 11.18 The approach presented in [70]. ANNs are trained to map the dis...

Figure 11.19 The scattering parameter

S

21

results (in dB) for a two-element ...

Figure 11.20 An illustration of the approach presented in [73]. The FDTD-bas...

Figure 11.21 The reported relative error in estimating the magnetic field

H

z

Figure 11.22 Results for the approach reported in [76]; The predicted real p...

Figure 11.23 The unsupervised learning approach reported in [80].

Chapter 12

Figure 12.1 MATLAB's parallel pool settings menu.

Figure 12.2 Selecting Cluster Profile Manager for additional setting.

Figure 12.3 The 3D representation of the microstrip low-pass filter problem....

Figure 12.4 Flowchart of parallelized FDTD MATLAB code using MATLAB's parall...

Figure 12.5 Boundaries of distributed sub-domains. The H-field components ar...

Figure 12.6 Comparison of S21 simulated values for microstrip low-pass filte...

Figure 12.7 Throughput in millions of cells per second for different numbers...

Figure 12.8 Throughput in millions of cells per second for different numbers...

Figure 12.9 Throughput in millions of cells per second for different numbers...

Figure 12.10 Throughput in millions of cells per second for different number...

Chapter 13

Figure 13.1 Treatment of overlapping polygons for interfaces between adjacen...

Figure 13.2 An example of the domain decomposition with different types of e...

Figure 13.3 Time overhead in 3000 steps with different DoFs (tet: tetrahedra...

Figure 13.4 Flowchart of the serial subdomain-level DGTD method and the para...

Figure 13.5 Repartition of the example in Figure 13.2. (a) Initial mesh and ...

Figure 13.6 Two partitions for the same case. (a) Less computational load di...

Figure 13.7 The mesh of the metallic sphere on the

y

–

z

plane (the coarse mes...

Figure 13.8 The bistatic RCS of the metallic sphere at 30 MHz including

E

-pl...

Figure 13.9 Geometry of the low-pass filter. (a) Overview; (b)

y

–

z

plane; (c...

Figure 13.10 Mesh of the low-pass filter with hexahedral elements.

Figure 13.11 The domain decomposition of the low-pass filter on the

x

–

y

plan...

Figure 13.12 Numerical results of the filter case from parallel subdomain-le...

Figure 13.13 The performance of the parallelization. (a) The ideal time over...

Figure 13.14 The overview of the desktop case. (a) The whole structure and (...

Figure 13.15 Adaptive partitioning process of (a) PML region and (b) physica...

Figure 13.16 The

x

component of E-field signals at the receivers of the desk...

Figure 13.17 The

x

components of the E-field inside and outside the desktop ...

Figure 13.18 The performance of the parallelization for the desktop case. (a...

Figure 13.19 Geometry of the patch antenna array. (a) Overview; (b) x − y pl...

Figure 13.20 Three representative scattering parameters of the patch antenna...

Figure 13.21 The partition of the patch antenna array case. (a) Initial part...

Figure 13.22 The performance of the parallelization for the patch antenna ar...

Figure 13.23 Geometry of the Boeing 737 plane. (a) Overview; (b)

y

–

z

plane; ...

Figure 13.24 RCS of Boeing 737 at 100 MHz in (a)

ϕ

direction and (b)

θ

...

Figure 13.25 The performance of the parallel computation for the Boeing 737 ...

Chapter 14

Figure 14.1 A graphical model of random variables shows the probabilistic de...

Figure 14.2 A factor graph representing the factorization of

in (14.10).

Figure 14.3 A FEM mesh (a) and its corresponding factor graph (b).

Figure 14.4 Breakdown of FETD computation time for an instantaneous nonlinea...

Figure 14.5 Depiction of non-coalesced versus coalesced GPU global memory ac...

Figure 14.6 GPU over CPU speedup as a function of degrees of freedom for bot...

Figure 14.7 The geometry of the test case in two dimensions.

Figure 14.8 Shared memory model.

Figure 14.9 Mesh coloring scheme for a structured quadrilateral mesh.

Figure 14.10 Comparison of temperature values over time at a specific point ...

Figure 14.11 Strong scaling of Algorithm 14.3 in terms of speedup with respe...

Figure 14.12 Comparison of Algorithm 14.3 versus PETSc execution times on 16...

Figure 14.13 The convergence plot of Algorithm 14.3 compared to that of a Ga...

Chapter 15

Figure 15.1 The propagation of the one-dimensional Gaussian pulse.

Figure 15.2 Schematic model of nanofilms.

Figure 15.3 A hybrid system of nanoantenna and quantum emitters.

Figure 15.4 The solving flow of quantum–EM coupled equation.

Figure 15.5 Space and time configurations of the field components (

E

,

H

,

A

,

Figure 15.6 The calculation flow of the coupled quantum–EM system.

Figure 15.7 A geometric model of a nanotube.

Figure 15.8 A procedure illustration of the proposed method for updating the...

Figure 15.9 The confining potential

V

along the

z

direction.

Figure 15.10 The electron wave packet distribution of the ground state and t...

Figure 15.11 The spatiotemporal plots of the probability density |

ψ

|

2

....

Figure 15.12 The magnitude of the transition rate factor Ω.

Figure 15.13 The simulation procedures of the symplectic algorithm for the c...

Figure 15.14 Boundary treatment of

A

field by the image theory.

Figure 15.15 (a) The particle is placed in a metal resonator. (b) The partic...

Figure 15.16 (a) The calculation results for a weak field (Ω = 0....

Figure 15.17 (a) The calculation results for a strong field (Ω = ...

Figure 15.18 (a) The calculation results for a small detuning Δ...

Figure 15.19 (a) The calculation results for a large detuning Δ...

Figure 15.20 The relative errors of the FDTD(2,2), FDTD(2,2)-DG, and SFDTD(4...

Figure 15.21 The population inversions of particles with or without quantum ...

Figure 15.22 The population inversions of particles with or without quantum ...

Figure 15.23 Changes of population inversion at different positions.

Figure 15.24 Changes of population inversion with different factors.

Chapter 16

Figure 16.1 A three-dimensional CFDTD computational space composed of...

Figure 16.2 Staggered arrangement of electromagnetic field components for a ...

Figure 16.3 The first layer of CFDTD cells around the

z

-axis are wedge-shape...

Figure 16.4 Field components at the vicinity of the

z

-axis.

Figure 16.5 Contour path for updating

E

z

(1,

j

,

k

)

.

Figure 16.6 Field components to update

Eϕ

in the PML region along the...

Figure 16.7 Field components to update

E

z

in the PML region along the

ρ

Figure 16.8 Three-dimensional CFDTD domain for testing radial CPML implement...

Figure 16.9 A cosine-modulated Gaussian waveform with a central frequency of...

Figure 16.10 CPML reflections for the first four rotational modes when the C...

Figure 16.11 CPML reflections for the first four rotational modes when the C...

Figure 16.12 CPML reflections for the first four rotational modes when the C...

Figure 16.13 CPML reflections for the first four rotational modes when the C...

Figure 16.14 A voltage source placed between nodes

(

i

,

j

,

k

)

and

(i, j, k + 

...

Figure 16.15 A voltage source placed between nodes

(

i

,

j

,

k

)

and

(i, j, k + 

...

Figure 16.16 Top view of the cylindrical FDTD space where the area of the su...

Figure 16.17 Top view of the cylindrical FDTD space where the area of the su...

Figure 16.18 A voltage source placed between nodes

(

i

,

j

,

k

)

and

(i, j, k + 

...

Figure 16.19 The voltage source places between nodes

(

i

,

j

,

k

)

and

(i, j, k 

...

Figure 16.20 A resistor placed between nodes

(

i

,

j

,

k

)

and

(i, j, k + 1)

...

Figure 16.21 A capacitor placed between nodes

(

i

,

j

,

k

)

and

(i, j, k + 1)

...

Figure 16.22 An inductor placed between nodes

(

i

,

j

,

k

)

and

(i, j, k + 1)

...

Figure 16.23 A simple circuit diagram in FDTD cell configuration where the e...

Figure 16.24 Sampled voltage across

Z

L

 = 

R

L

from the CFDTD sim...

Figure 16.25 A voltage source with

ZS = RS + ZC

...

Figure 16.26 Sampled voltage across

Z

L

 = 

R

L

from the CFDTD sim...

Figure 16.27 A circuit configuration where the elements are placed in the

φ

...

Figure 16.28 The voltage across the resistive load

R

L

in the circuit in Figu...

Figure 16.29 A circuit configuration in CFDTD where the elements are placed ...

Figure 16.30 Voltage across

R

L

for the elements placed in the

ρ

directi...

Figure 16.31 Circuit configuration of a source connected to two parallel loa...

Figure 16.32 A FDTD circuit diagram for the circuit in Figure 16.31.

Figure 16.33 Voltage across the two parallel loads in Figure 16.32.

Figure 16.34 Circuit configuration of parallel sources and parallel resistor...

Figure 16.35 A FDTD circuit representation with parallel sources and paralle...

Figure 16.36 Voltage across the two loads in Figure 16.35.

Figure 16.37 Time domain waveform and sampled voltage across the voltage sou...

Figure 16.38 Frequency domain sampled voltage and analytical solution across...

Figure 16.39 The absolute error between sampled voltage and the theoretical ...

Guide

Cover

Table of Contents

Title Page

Copyright

About the Editors

List of Contributors

Preface

Begin Reading

Index

End User License Agreement

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IEEE Press

445 Hoes Lane

Piscataway, NJ 08854

IEEE Press Editorial Board

Sarah Spurgeon, Editor in Chief

Jón Atli Benediktsson

Anjan Bose

Adam Drobot

Peter (Yong) Lian

Andreas Molisch

Saeid Nahavandi

Jeffrey Reed

Thomas Robertazzi

Diomidis Spinellis

Ahmet Murat Tekalp

Advances in Time-Domain Computational Electromagnetic Methods

Edited by Qiang Ren, Su Yan, and Atef Z. Elsherbeni

IEEE Press Series on Electromagnetic Wave Theory

Copyright © 2023 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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About the Editors

Qiang Ren (S'12–M'16–SM'22) received both the B.S. and M.S. degrees in electrical engineering from Beihang University, Beijing, China, and Institute of Acoustics, Chinese Academy of Sciences, Beijing, China, in 2008 and 2011, respectively, and the PhD degree in Electrical Engineering from Duke University, Durham, NC, in 2015. From 2016 to 2017, he was a postdoctoral researcher with the Computational Electromagnetics and Antennas Research Laboratory (CEARL) of the Pennsylvania State University, University Park, PA. In Sept 2017, he joined the School of Electronics and Information Engineering, Beihang University, Beijing, China, as an “Excellent Hundred” Associate Professor.

Dr. Ren is the recipient of The Applied Computational Electromagnetics Society (ACES) Early Career Award in 2021, the Young Scientist Award of APEMC 2022, and the Young Scientist Award of 2018 International Applied Computational Electromagnetics Society (ACES) Symposium in China. He serves as the Associate Editor of IEEE Journal of Multiscale and Multiphysics Simulation Techniques, ACES Journal, and Microwave and Optical Technology Letters (MOTL), and also serves as a reviewer for more than 30 journals. He has published more than 90 papers in peer-reviewed journals and conferences. The students he advised have received multiple awards including Best Student Paper 3rd Prize of PIERS 2021 and Best Paper Shortlist of CSRSWTC 2021, and Honorable Mention Award of AP-S URSI 2022. His current research interests include numerical modeling methods for complex media, multiscale and multiphysics problems, inverse scattering, deep learning, and parallel computing.

Dr. Su Yan (S'08-M'12-SM'17) received the B.S. degree in electromagnetics and microwave technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2005, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana–Champaign (UIUC), Urbana, IL, USA, in 2012 and 2016, respectively. He joined the Howard University, Washington, DC, USA, as an Assistant Professor in 2018, and serves as the Director of Graduate Studies in the Department of Electrical Engineering and Computer Science since 2020. He has authored or coauthored over 110 papers in refereed journals. His current research interests include nonlinear electromagnetic and multiphysics problems, electromagnetic scattering and radiation, numerical methods in computational electromagnetics, especially continuous and discontinuous Galerkin finite element methods, integral-equation-based methods, domain decomposition methods, fast algorithms, and preconditioning techniques.

Dr. Yan is a Senior Member of the Institute of Electrical and Electronics Engineers (IEEE) and a Life Member of the Applied Computational Electromagnetics Society (ACES). He was a recipient of the ACES Early Career Award “for contributions to linear and nonlinear electromagnetic and multiphysics modeling and simulation methods" in 2020, the P. D. Coleman Outstanding Research Award, and the Yuen T. Lo Outstanding Research Award by the Department of Electrical and Computer Engineering, UIUC, in 2015 and 2014, respectively. He was also a recipient of the Edward E. Altschuler AP-S Magazine Prize Paper Award by IEEE Antennas and Propagation Society in 2020, the Best Student Paper Award (the first place winner) at the IEEE ICWITS/ACES 2016 Conference, Honolulu, HI, USA, in 2016, the USNC/URSI Travel Fellowship Grant Award by the National Academies in 2015, the Best Student Paper Award (the first place winner) at the 27th International Review of Progress in ACES, Williamsburg, VA, USA, in 2011, and the Best Student Paper Award by the IEEE Chengdu Section in 2010. He serves as an Associate Editor for IEEE Access, an Associate Editor for the International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, and a reviewer for multiple journals and conferences.

Atef Z. Elsherbeni received an honor's B.Sc. degree in Electronics and Communications, an honor's B.Sc. degree in Applied Physics, and a M.Eng. degree in Electrical Engineering, all from Cairo University, Cairo, Egypt, in 1976, 1979, and 1982, respectively, and a Ph.D. degree in Electrical Engineering from Manitoba University, Winnipeg, Manitoba, Canada, in 1987. He started his engineering career as a part-time Software and System Design Engineer from March 1980 to December 1982 at the Automated Data System Center, Cairo, Egypt. From January to August 1987, he was a Post-Doctoral Fellow at Manitoba University. Dr. Elsherbeni joined the faculty at the University of Mississippi in August 1987 as an Assistant Professor of Electrical Engineering. He advanced to the rank of Associate Professor in July 1991 and to the rank of Professor in July 1997. He was the Associate Dean of the College of Engineering for Research and Graduate Programs from July 2009 to July 2013 at the University of Mississippi. He then joined the Electrical Engineering and Computer Science (EECS) Department at Colorado School of Mines in August 2013 as the Dobelman Distinguished Chair Professor. He was appointed the Interim Department Head for (EECS) from 2015 to 2016 and from 2016 to 2018 he was the Electrical Engineering Department Head. He spent a sabbatical term in 1996 at the Electrical Engineering Department, University of California at Los Angeles (UCLA), and was a Visiting Professor at Magdeburg University during the summer of 2005 and at Tampere University of Technology in Finland during the summer of 2007. In 2009, he was selected as Finland Distinguished Professor by the Academy of Finland and TEKES.

Over the years, Dr. Elsherbeni participated in acquiring millions of dollars to support his research group activities dealing with scattering and diffraction of EM waves by dielectric and metal objects, finite difference time domain analysis of antennas and microwave devices, field visualization and software development for EM education, interactions of electromagnetic waves with human body, RFID and sensor integrated FRID systems, reflector and printed antennas and antenna arrays for radars, UAV, and personal communication systems, antennas for wideband applications, and measurements of antenna characteristics and material properties. Dr. Elsherbeni is an IEEE life fellow and ACES fellow. He is the Editor-in-Chief for ACES Journal and a past Associate Editor of the Radio Science Journal. He was the Chair of the Engineering and Physics Division of the Mississippi Academy of Science, the Chair of the Educational Activity Committee for IEEE Region 3 Section, and the general Chair for the 2014 APS-URSI Symposium and the president of ACES Society from 2013 to 2015. Dr. Elsherbeni is selected as Distinguished Lecturer for IEEE Antennas and Propagation Society for 2020-2023.

List of Contributors

David S. Abraham

Department of Electrical and Computer Engineering

McGill University

Montréal, Québec

Canada

Amir Akbari

Department of Electrical & Computer Engineering

McGill University

Montréal, Québec

Canada

Ali Akbarzadeh-Sharbaf

Department of Electrical and Computer Engineering

McGill University

Montréal, Québec

Canada

Abdullah Algarni

Department of Electrical Engineering

King Fahd University of Petroleum and Minerals, Dhahran

Saudi Arabia

Shirook Ali

Faculty of Applied Science and Technology

Sheridan College

Brampton, Ontario

Canada

Francesco P. Andriulli

Department of Electronics

Politecnico di Torino

Torino

Italy

Hakan Bagci

Electrical and Computer Engineering (ECE) Program

Division of Computer, Electrical, and Mathematical

Science and Engineering (CEMSE)

King Abdullah University of Science and Technology (KAUST)

Thuwal

Saudi Arabia

Mohamed H. Bakr

Department of Electrical and Computer Engineering

McMaster University

Hamilton, Ontario

Canada

Jiefu Chen

Department of Electrical and Computer Engineering

University of Houston

Houston

United States

Liang Chen

Electrical and Computer Engineering (ECE) Program, Division of Computer, Electrical, and Mathematical Science and Engineering (CEMSE)

King Abdullah University of Science and Technology (KAUST)

Thuwal

Saudi Arabia

Rui Chen

Division of Computer, Electrical, and Mathematical Science and Engineering (CEMSE)

King Abdullah University of Science and Technology (KAUST)

Thuwal

Saudi Arabia

Xibi Chen

Tsinghua University

Beijing

China

Kristof Cools

Department of Information Tech.,

Ghent University

Belgium

Alexandre Dély

Department of Electronics

Politecnico di Torino

Torino

Italy

Veysel Demir

Electrical Engineering Department

Northern Illinois University

Chicago, IL

USA

Ming Dong

Electrical and Computer Engineering (ECE) Program, Division of Computer, Electrical, and Mathematical Science and Engineering (CEMSE)

King Abdullah University of Science and Technology (KAUST)

Thuwal

Saudi Arabia

Atef Z. Elsherbeni

Electrical Engineering Department

Colorado School of Mines

Golden, CO

USA

Dennis D. Giannacopoulos

Department of Electrical and Computer Engineering

McGill University

Montréal, Québec

Canada

Mohammed Hadi

Electrical Engineering Department

Colorado School of Mines

Golden, CO

USA

Yanyan Hu

Department of Electrical and Computer Engineering

University of Houston

Houston

USA

Yuyang Hu

Department of Electronic Engineering

Shanghai Jiao Tong University

Shanghai

P.R. China

Shifeng Huang

Department of Electronic Engineering

Shanghai Jiao Tong University

Shanghai

P.R. China

Zhixiang Huang

Key Laboratory of Intelligent Computing and Signal Processing

Ministry of Education

Anhui University

Hefei

China

Yunfeng Jia

Research Institute of Frontier Science

Beihang University

Beijing

China

Lijun Jiang

Department of Electrical and Electronic Engineering

The University of Hongkong

Hongkong

China

Yuchen Jin

Department of Electrical and Computer Engineering

University of Houston

Houston

USA

Joshua M. Kast

Colorado School of Mines

Electrical Engineering Department

Golden, CO

USA

Chao Li

School of Electronics and Information Engineering

Beihang University

Beijing

China

Ping Li

Department of Electrical Engineering

Shanghai Jiao Tong University

Shanghai

China

Rui Liu

Department of Electronic Engineering

Shanghai Jiao Tong University

Shanghai

P.R. China

Adrien Merlini

Microwave department

IMT Atlantique

Brest

France

Jiamei Mi

School of Electronics and Information Engineering

Beihang University

Beijing

China

Qiang Ren

School of Electronics and Information Engineering

Beihang University

Beijing

China

Xingang Ren

Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education

Anhui University

Hefei

China

Sadeed B. Sayed

School of Electrical and Electronic Engineering

Nanyang Technological University

Singapore

Wei E.I. Sha

Key Laboratory of Micro-Nano Electronic Devices and Smart Systems of Zhejiang Province

College of Information Science and Electronic Engineering

Zhejiang University

Hangzhou

China

Xuezhe Tian

Department of Electronic Engineering

Shanghai Jiao Tong University

Shanghai

P.R. China

Huseyin A. Ulku

Department of Electronics Engineering

Gebze Technical University

Kocaeli

Turkey

Alec Weiss

Electrical Engineering Department

Colorado School of Mines

Golden, CO

USA

Pengfei Wen

School of Electronics and Information Engineering

Beihang University

Beijing

China

Kaiming Wu

School of Electronics and Information Engineering

Beihang University

Beijing

China

Xuqing Wu

Department of Information and Logistics Technology

University of Houston

Houston

USA

Gaobiao Xiao

Department of Electronic Engineering

Shanghai Jiao Tong University

Shanghai

P.R. China

Guoda Xie

Key Laboratory of Intelligent Computing and Signal Processing

Ministry of Education

Anhui University

Hefei

China

Su Yan

Department of Electrical Engineering and Computer Science

Howard University

Washington, DC

USA

Fan Yang

Tsinghua University

Beijing

China

Wei Zhang

Sino French Engineering School

Beihang University

Beijing

China

Preface

Computational electromagnetics (CEM) research aims at the modeling and simulation of scientific and engineering problems based on the solution of Maxwell's equations or their variations through the development of numerical algorithms and computer programs for the evaluation, prediction, and optimization purposes. Since their early developments in 1960s, CEM methods have been used in widespread areas, such as radar scattering evaluation, antennas and array design, microwave circuit analysis, electronics and nanodevice development, magnetic and electric machine modeling, high-power microwave system simulation, bioelectromagnetic effect and biomedical device modeling, and electromagnetic imaging and inversion, just to name a few.

Through the past 70 some years, CEM methods have evolved into two major categories, asymptotic and full-wave methods. The asymptotic methods are based on an optical description of electromagnetic waves at high frequencies when the problem is electrically large. This category includes the basic geometrical optics (GO) and physical optics (PO) methods, which are later extended with the theory of diffraction to obtain the methods based on the geometrical theory of diffraction (GTD) and the physical theory of diffraction (PTD). Later developments include the unified theory of diffraction (UTD) and the shooting and bouncing ray (SBR) methods. These asymptotic methods, in general, are very efficient in the simulation of electromagnetic problems due to their simplified description of electromagnetic waves and are accurate only in an asymptotic sense, which means their accuracy is good only when the operating frequency is very high, or the problem size is very large compared to the wavelength. As a result, the asymptotic methods are usually regarded as “high-frequency methods.”

Another type of methods, known as the full-wave methods, are based on the rigorous numerical solution of Maxwell's equations, the Helmholtz equation, or various integral equations, in either the frequency or the time domain. Two types of numerical methods have been developed based on the mathematical nature of the governing equations, including the ones that solve partial differential equations (PDEs) and those that solve integral equations (IEs). Well-known PDE solvers are the frequency-domain finite difference method (FDM), finite-element method (FEM), and their time-domain counterparts, the finite-difference time-domain (FDTD) method, and the finite-element time-domain (FETD) method. On the IE side, the method of moments (MoM) has been widely used to solve surface integral equations (SIEs), volume integral equations (VIEs), and their combination, the volume–surface integral equations (VSIEs). The FDM and FDTD are very popular in electromagnetic and optical modeling and simulations due to their simplicity in formulation and implementation. But they generally suffer from the low geometrical modeling accuracy due to the use of structured meshes, the low spatial and temporal interpolation accuracy due to the use of finite differencing that is usually second-order accurate, and a large number of time steps due to the use of conditionally stable time integration schemes. Compared to FDM and FDTD, the FEM and FETD are very flexible and accurate in describing complex geometrical structures with unstructured conformal meshes, convenient in achieving higher-order accuracy with high-order interpolatory or hierarchical basis functions, and efficient in time integration with unconditionally stabile time marching schemes in the temporal discretization. However, although FEM and FETD convert PDEs into sparse matrix equations that have a linear storage complexity, the dimension of the matrix equations is usually very large since the simulation domains need to be discretized into volumetric meshes and consequently, the numerical solution of the matrix equations can be very time consuming. Based on the solution of IEs, the MoM only requires the discretization of either the surface or the volume of the objects without the need of modeling their surrounding background or the truncation boundary. As a result, the overall dimension of the matrix equations is much smaller compared to that of the same problem modeled by FEM. Unfortunately, due to the use of Green's function that depicts the global coupling of fields between every two points in the objects, the resulting system matrix is a fully populated dense matrix that requires an storage with N being the number of degrees of freedoms (DoFs) and either or solution cost with a direct or an iterative solver, respectively. This greatly limits the size of the problem that a full-wave method can handle. Since 1990s, various fast algorithms have been developed to reduce the storage and computational complexities, and domain decomposition methods based on the philosophy of “divide and conquer” have been developed. The further application of large-scale parallel computation boosted the modeling and simulation capabilities of the modern CEM methods significantly. Many EM problems that cannot be tackled by the full-wave methods in the past can now be solved with high accuracy and good efficiency.

Full-wave simulations can be performed in both the frequency and the time domains. The numerical solutions obtained from these two types of methods are related by Fourier/inverse Fourier transform. When performing wideband or transient simulations, the frequency- and time-domain methods are equivalent. On the one hand, frequency-domain simulations can be performed on multiple frequencies of interest and their solutions can be inverse Fourier transformed to the time domain to construct time-domain results. On the other hand, time-domain simulations can be performed using a transient excitation to calculate the time-domain solutions, which can then be Fourier transformed to the frequency domain and sampled at the frequencies of interest. Apparently, in solving ultra-wideband problems, the time-domain simulation will outperform its frequency-domain counterpart since the excitation in these applications only lasts for a very short period but will lead to a very large number of sampling frequencies and a very long simulation time when employing frequency-domain solvers. There are, in fact, many other scenarios where time-domain simulations are not only preferred but, many times, required. Typical examples include the simulation of time-modulated materials, nonlinear problems, and multiphysics problems.